Etymonline has an informative entry on the etymology of anoint:
c.1300 (implied in anointing), from O.Fr. enoint "smeared on," pp. of enoindre "smear on," from L. inunguere "to anoint," from in- "on" + unguere "to smear" (see unguent). Originally in reference to grease or oil smeared on for medicinal purposes; its use in the Coverdale Bible in reference to Christ (cf. The Lord's Anointed, see chrism) has spiritualized the word.
ODO's entry for anoint lists three senses:
- (anoint something with) smear or rub something with (any other substance):
Kuna Indians anoint the tips of their arrows with poison
- ceremonially confer divine or holy office upon (a priest or monarch) by smearing or rubbing with oil:
[with object and complement]:
Samuel anointed him king
- nominate or choose (someone) as successor to or leading candidate for a position:
(as adjective anointed)
his officially anointed heir
Considering the dearth of royalty in the modern world, it's easy to see how the word has evolved to apply to political and industrial leaders. Furthermore, the "crowning" of the new leader is often performed in a ceremony making the term quite fitting. In the meantime, religious leaders and royalty continue to be anointed as before.
While anoint started its life innocently enough, it's since been hijacked and carries, if not always religious, ceremonial connotations to this day.